aa r X i v : . [ m a t h . QA ] J a n DEFORMED FOURIER MODELS WITH FORMAL PARAMETERS
TEODOR BANICA
Abstract.
The deformed Fourier matrices H = F M ⊗ Q F N , with Q ∈ T MN , producea matrix model C ( S + MN ) → M MN ( C ( T MN )). When Q ∈ T MN is generic, the corre-sponding fiber can be investigated via algebraic techniques, and the main character lawis asymptotically free Poisson. We present here an alternative point of view on thesequestions, using formal parameters instead of generic parameters, and analytic tools. Introduction
It is well-known that the unitary representations of a discrete group Γ are in one-to-onecorrespondence with the representations of the group algebra C ∗ (Γ). Now given a discretesubgroup Γ ⊂ U N , we obtain a representation π : C ∗ (Γ) → M N ( C ). This representationis in general not faithul, its target algebra being finite dimensional. On the other hand,this representation “reminds” Γ. We say that π is inner faithful.The inner faithful representations can be in fact axiomatized in the general discretequantum group context. Given such a quantum group Γ, and a representation π : C ∗ (Γ) → B , one can construct a biggest quotient Γ → Λ producing a factorization π : C ∗ (Γ) → C ∗ (Λ) → B , and π is called inner faithful when Γ = Λ. See [3].This construction is of particular interest when formulated from a dual viewpoint, withΓ = b G , and with B = M K ( C ( X )) being a random matrix algebra. To be more precise,given a compact quantum group G , and a matrix model π : C ( G ) → M K ( C ( X )), onecan construct a biggest closed subgroup H ⊂ G producing a factorization π : C ( G ) → C ( H ) → M K ( C ( X )), and π is called inner faithful when G = H . See [3].Generally speaking, an inner faithful model π : C ( G ) → M K ( C ( X )) can be regarded asbeing a source of interesting information about G , of both algebraic and analytic nature.Thus, we have here a new method for investigating the compact quantum groups. Thismethod is alternative to the pure algebraic geometric point of view (“easiness”).A number of tools for dealing with the inner faithful models have been developed,some of them being algebraic [3], [4], [10], [12], and some other, analytic [6], [11], [20],[24]. However, at the level of concrete examples, only a few models have been succesfullyinvestigated, so far. Among them is the model C ( S + MN ) → M MN ( C ) coming from adeformed Fourier matrix H = F M ⊗ Q F N , with parameter Q ∈ T MN . Mathematics Subject Classification.
Key words and phrases.
Quantum permutation, Matrix model.
The story with these latter models is long and twisted, and involved many people. Asa brief summary, the development of the subject was as follows:(1) Given an arbitrary inner faithful model C ( G ) → M K ( C ), an abstract formula forthe Haar integration over G , based on [15], was found in [6].(2) The representations C ( S + MN ) → M MN ( C ) coming from deformed Fourier matriceswith generic parameters were studied in [4], using algebraic techniques.(3) Some applications of the integration formula in [6], to the deformed Fourier matrixrepresentations, were found short afterwards, in [2].(4) In the meantime, the algebraic methods in [4] were substantially extended, as tocover certain non-generic parameters Q ∈ T MN , in [10].(5) In the meantime as well, a generalization of the integration formula in [6], coveringthe models C ( G ) → M K ( C ( X )), was found in [24].(6) The integration formula in [24] was applied to certain related representations, oftype C ( S + N ) → M N ( C ( U N )), in the recent paper [7].The purpose of this paper is to study the deformed Fourier models, using analytictechniques. We will take advantage of the recent formula in [24], and investigate the fullparametric model π : C ( S + MN ) → M MN ( C ( T MN )), instead of its individual fibers. Theformula in [24] will turn to apply well, and to lead to concrete results. As in [4], our mainresult will state that main character becomes free Poisson, in the M = tN → ∞ limit.We will discuss as well a number of further properties of the main character.These results can be deduced as well from [4], since in the probabilistic picture for themoments, the non-generic parameters do not count. However, we believe that having afully analytic proof is a good thing. In short, following [7], we have now a second concreteapplication of the integration formula in [6], [24]. Our hope is that this formula can beapplied to some other situations, and could eventually become a serious alternative to theWeingarten formula [5], [13], and to the “easiness” methods in general [8], [18].The paper is organized as follows: 1-2 are preliminary sections, in 3-4 we study thetruncated moments of the main character, in 5-6 we compute the plain moments of themain character, in 7-8 we work out a number of moment estimates, and in 9-10 we stateand prove our main results, and we end with a few concluding remarks.1. Quantum groups
We use the quantum group formalism of Woronowicz [25], [26], with the extra axiom S = id . That is, we consider pairs ( A, u ) consisting of a C ∗ -algebra A , and a unitarymatrix u ∈ M N ( A ), such that the following formulae define morphisms of C ∗ -algebras:∆( u ij ) = X k u ik ⊗ u kj , ε ( u ij ) = δ ij , S ( u ij ) = u ∗ ji These morphisms are called comultiplication, counit and antipode. The abstract spec-tum G = Spec ( A ) is called compact quantum group, and we write A = C ( G ). EFORMED FOURIER MODELS WITH FORMAL PARAMETERS 3
The example that we are interested in, due to Wang [23], is as follows:
Definition 1.1. C ( S + N ) is the universal C ∗ -algebra generated by the entries of a N × N matrix u = ( u ij ) which is magic, in the sense that its entries are projections ( p = p ∗ = p ),summing up to on each row and each column of u . This algebra satisfies Woronowicz’s axioms, and the underlying noncommutative space S + N is therefore a quantum group, called quantum permutation group. We have an inclu-sion S N ⊂ S + N , which is an isomorphism at N = 1 , ,
3, but not at N ≥
4. See [23].Now back to the general case, we have the following key notion, fom [3]:
Definition 1.2.
Let π : C ( G ) → M K ( C ( T )) be a C ∗ -algebra representation. (1) The Hopf image of π is the smallest quotient Hopf C ∗ -algebra C ( G ) → C ( H ) producing a factorization of type π : C ( G ) → C ( H ) → M K ( C ( T )) . (2) When the inclusion H ⊂ G is an isomorphism, i.e. when there is no non-trivialfactorization as above, we say that π is inner faithful. As a basic example, when G = b Γ is a group dual, π must come from a group repre-sentation Γ → C ( T, U K ), and the factorization in (1) is the one obtained by taking theimage, Γ → Γ ′ ⊂ C ( T, U K ). Thus π is inner faithful when Γ ⊂ C ( T, U K ).Also, given a compact group G , and elements g , . . . , g K ∈ G , we can consider therepresentation π = ⊕ i ev g i : C ( G ) → C K . The minimal factorization of π is then via C ( G ′ ), with G ′ = < g , . . . , g K > . Thus π is inner faithful when G = < g , . . . , g K > .We recall that an Hadamard matrix is a square matrix H ∈ M N ( C ) whose entries areon the unit circle, and whose rows are pairwise orthogonal. Given a parametric family ofsuch matrices, { H x | x ∈ T } , we can consider the corresponding element H ∈ M N ( C ( T )),that we call as well Hadamard matrix. The relation with S + N comes from: Definition 1.3.
Associated to H ∈ M N ( C ( T )) Hadamard is the representation π : C ( S + N ) → M N ( C ( T )) , π ( u ij ) : x → P roj ( H xi /H xj ) where H x , . . . , H xN ∈ T N are the rows of H x , and the quotients are taken inside T N . Here the fact that the projections on the right form a magic matrix, and hence producea representation of C ( S + N ), follows from the Hadamard matrix condition.The problem is that of computing the Hopf image of the above representation. Thereis only one basic example here, namely the one coming from the Fourier coupling F G ∈ M G × b G ( C ) of a finite abelian group G . Here the representation constructed above factorizesas π : C ( S + G ) → C ( S G ) → C ( G ) → M N ( C ), and the Hopf image is C ( G ).In order to approach the problem, we use tools from [6], [24]. Let us first go back tothe general context of Definition 1.2, and assume that T is a measured space, so that wehave a trace tr : M K ( C ( T )) → C , given by tr ( M ) = K P Ki =1 R X M ii ( x ) dx .We have then the following key result, from [6], [24]: TEODOR BANICA
Proposition 1.4.
Given an inner faithful model π : C ( G ) → M K ( C ( T )) , we have Z G = lim k →∞ k k X r =1 ( tr ◦ π ) ∗ r in moments, with the convolutions at right being given by φ ∗ ψ = ( φ ⊗ ψ )∆ .Proof. This was proved in [6] in the case X = { . } , using theory from [15], the ideabeing that the Haar state can be obtained by starting with an arbitrary positive linearfunctional, and then convolving. The general case was established in [24]. (cid:3) In the case where G has a fundamental corepresentation u = ( u ij ), the above result hasa more concrete formulation, of linear algebra flavor, as follows: Proposition 1.5.
Given an inner faithful model π : C ( G ) → M K ( C ( T )) , mapping u ij → U ij , the moments of χ = P i u ii with respect to R rG = ( tr ⊗ π ) ∗ r are the numbers c rp = T r ( T rp ) : ( T p ) i ...i p ,j ...j p = tr ( U i j . . . U i p j p ) and these numbers converge with r → ∞ to the moments of χ with respect to R G .Proof. By evaluating R rG = ( tr ⊗ π ) ∗ r on a product of coefficients, we obtain: Z rG u i j . . . u i p j p = ( T rp ) i ...i p ,j ...j p Now by summing over i x = j x , this gives the formula in the statement. See [6]. (cid:3) We can apply Proposition 1.5 to the Hadamard representations, and we obtain:
Theorem 1.6.
For the representation coming from H ∈ M N ( C ( T )) we have c rp = 1 N ( p +1) r Z T r X i ...i rp X j ...j rp H x i j H x i j H x i j H x i j . . . H x i p j p H x i p j H x i p j H x i p j p . . . . . . H x r i r j r H x r i j r H x r i r j r H x r i j r . . . H x r i rp j rp H x r i p j r H x r i rp j r H x r i p j rp dx and these numbers converge with r → ∞ to the moments of χ with respect to R G .Proof. We have indeed the following computation: c rp = X i ...i rp ( T p ) i ...i p ,i ...i p . . . . . . ( T p ) i r ...i rp ,i ...i p = Z T r X i ...i rp tr ( U x i i . . . U x i p i p ) . . . . . . tr ( U x r i r i . . . U x r i rp i p ) dx = 1 N r Z T r X i ...i rp X j ...j rp ( U x i i ) j j . . . ( U x i p i p ) j p j . . . . . . ( U x r i r i ) j r j r . . . ( U x r i rp i p ) j rp j r dx In terms of H , this gives the formula in the statement. See [2]. (cid:3) EFORMED FOURIER MODELS WITH FORMAL PARAMETERS 5 Fourier models
As mentioned in section 1, the “simplest” matrix model is the one coming from theFourier matrix F G ∈ M G × b G ( C ) of a finite abelian group G , where the associated quantumgroup is G itself. Our purpose here will be that of investigating the “next simplest”models. These appear by deforming the Fourier matrices, or rather the tensor productsof such matrices, F G × H = F G ⊗ F H , via the following construction, due to Dit¸˘a [14]: Proposition 2.1.
The matrix F G × H ∈ M G × H ( T G × H ) given by ( F G × H ) ia,jb ( Q ) = Q ib ( F G ) ij ( F H ) ab is complex Hadamard, and its fiber at Q = (1 ib ) is the Fourier matrix F G × H .Proof. The fact that the rows of F G ⊗ Q F H = F G × H ( Q ) are pairwise orthogonal followsfrom definitions, see [14]. With 1 = (1 ij ) we have ( F G ⊗ F H ) ia,jb = ( F G ) ij ( F H ) ab , and werecognize here the formula of F G × H = F G ⊗ F H , in double index notation. (cid:3) The fibers F G ⊗ Q F H = F G × H ( Q ) were investigated in [4], and then in [10], by usingalgebraic techniques. Our purpose here is that of obtaining some related results, regardingthe matrix F G × H itself, by using analytic techniques. We have: Theorem 2.2.
For the representation coming from F G × H we have c rp = 1 M r +1 N i , . . . , i r , a , . . . , a p ∈ { , . . . , M − } ,b , . . . , b p ∈ { , . . . , N − } , [( i x + a y , b y ) , ( i x +1 + a y , b y +1 ) | y = 1 , . . . , p ]= [( i x + a y , b y +1 ) , ( i x +1 + a y , b y ) | y = 1 , . . . , p ] , ∀ x where M = | G | , N = | H | , and the sets between brackets are sets with repetitions.Proof. We use the formula in Theorem 1.6. With K = F G , L = F H we have: c rp = 1( M N ) r Z T r X i ...i rp X b ...b rp Q i b Q i b Q i b Q i b . . . Q i p b p Q i p b Q i p b Q i p b p . . . . . . Q ri r b r Q ri b r Q ri r b r Q ri b r . . . Q ri rp b rp Q ri p b r Q ri rp b r Q ri p b rp M pr X j ...j rp K i j K i j K i j K i j . . . K i p j p K i p j K i p j K i p j p . . . . . . K i r j r K i j r K i r j r K i j r . . . K i rp j rp K i p j r K i rp j r K i p j rp N pr X a ...a rp L a b L a b L a b L a b . . . L a p b p L a p b L a p b L a p b p . . . . . . L a r b r L a b r L a r b r L a b r . . . L a rp b rp L a p b r L a rp b r L a p b rp dQ TEODOR BANICA
Since we are in the Fourier matrix case, K = F G , L = F H , we can perform the sumsover j, a . To be more precise, the last two averages appearing above are respectively:∆( i ) = Y x Y y δ ( i xy + i x +1 y − , i x +1 y + i xy − )∆( b ) = Y x Y y δ ( b xy + b x +1 y − , b x +1 y + b xy − )We therefore obtain the following formula for the truncated moments of the maincharacter, where ∆ is the product of Kronecker symbols constructed above: c rp = 1( M N ) r Z T r X ∆( i )=∆( b )=1 Q i b Q i b Q i b Q i b . . . Q i p b p Q i p b Q i p b Q i p b p . . . . . . Q ri r b r Q ri b r Q ri r b r Q ri b r . . . Q ri rp b rp Q ri p b r Q ri rp b r Q ri p b rp dQ Now by integrating with respect to Q ∈ ( T G × H ) r , we are led to counting the multi-indices i, b satisfying the condition ∆( i ) = ∆( b ) = 1, along with the following conditions,where the sets between brackets are by definition sets with repetitions: (cid:2) i b . . . i p b p i b . . . i p b (cid:3) = (cid:2) i b . . . i p b i b . . . i p b p (cid:3) ... (cid:2) i r b r . . . i rp b rp i b r . . . i p b r (cid:3) = (cid:2) i r b r . . . i rp b r i b r . . . i p b rp (cid:3) In a more compact notation, the moment formula is therefore as follows: c rp = 1( M N ) r n i, b (cid:12)(cid:12)(cid:12) ∆( i ) = ∆( b ) = 1 , [ i xy b xy , i x +1 y b xy +1 ] = [ i xy b xy +1 , i x +1 y b xy ] , ∀ x o Now observe that the above Kronecker type conditions ∆( i ) = ∆( b ) = 1 tell us thatthe arrays of indices i = ( i xy ) , b = ( b xy ) must be of the following special form: i . . . i p . . .i r . . . i rp = i + a . . . i + a p . . .i r + a . . . i r + a p , b . . . b p . . .b r . . . b rp = j + b . . . j + b p . . .j r + b . . . j r + b p Here all the new indices i x , j x , a y , b y are uniquely determined, up to a choice of i , j .Now by replacing i xy , b xy with these new indices i x , j x , a y , b y , with a M N factor added,which accounts for the choice of i , j , we obtain the following formula: c rp = 1( M N ) r +1 (cid:26) i, j, a, b (cid:12)(cid:12)(cid:12) [( i x + a y , j x + b y ) , ( i x +1 + a y , j x + b y +1 )]= [( i x + a y , j x + b y +1 ) , ( i x +1 + a y , j x + b y )] , ∀ x (cid:27) Now observe that we can delete if we want the j x indices, which are irrelevant. Thus,we obtain the formula in the statement. (cid:3) Summarizing, the Haar integration formula in [24] leads to a combinatorial interpre-tation of the moments of the main character. In what follows we will investigate thesemoments, first with some exact computations, and then with analytic techniques.
EFORMED FOURIER MODELS WITH FORMAL PARAMETERS 7 Exact computations
In this section and in the next one we study the numbers c rp found in Theorem 2.2,with a number of exact computations. Observe first that these numbers depend only on M = | G | and N = | H | . We denote in what follows these numbers by c rp ( M, N ).As an illustration, here are a few trivial computations:
Proposition 3.1.
The numbers c rp ( M, N ) have the following properties: (1) c rp (1 , N ) = N p − . (2) c rp ( M,
1) = M p − . (3) c r ( M, N ) = 1 . (4) c p ( M, N ) = (
M N ) p − .Proof. In all the cases under investigation, the conditions on the sets with repetitions inTheorem 2.2 are trivially satisfied, and this gives the above formulae. (cid:3)
We have in fact the following result, including all the “obvious” information:
Proposition 3.2.
The following normalized quantities belong to [0 , , d rp ( M, N ) = 1(
M N ) p − · c rp ( M, N ) and are equal to at M = 1 , N = 1 , p = 1 or r = 1 .Proof. According to Theorem 2.2, the rescaled moments are given by: d rp ( M, N ) = 1 M p + r N p i , . . . , i r , a , . . . , a p ∈ { , . . . , M − } ,b , . . . , b p ∈ { , . . . , N − } , [( i x + a y , b y ) , ( i x +1 + a y , b y +1 )]= [( i x + a y , b y +1 ) , ( i x +1 + a y , b y )] , ∀ x Thus d rp ( M, N ) ∈ [0 , (cid:3) Let us perform now some computations. The formulae look better for the numbers d rp ( M, N ) in Proposition 3.2, so we will use these numbers. First, we have:
Proposition 3.3.
When one of i, a, b consists of equal indices, the conditions defining d rp ( M, N ) are trivially satisfied. The corresponding contribution is α rp ( M, N ) = 1 − ( M p − M )( M r − M )( N p − N ) M p + r N p and this quantity equals d rp ( M, N ) at M = 1 , N = 1 , r = 1 , or p ≤ . TEODOR BANICA
Proof.
Assume that one of i, a, b consists of equal indices. By translation we can assumethat this common index is 0, and the conditions defining d rp ( M, N ) read: i x = 0 : [( a y , b y ) , ( a y , b y +1 )] = [( a y , b y +1 ) , ( a y , b y )] a y = 0 : [( i x , b y ) , ( i x +1 , b y +1 )] = [( i x , b y +1 ) , ( i x +1 , b y )] b y = 0 : [( i x + a y , , ( i x +1 + a y , i x + a y , , ( i x +1 + a y , i x = 0 or b y = 0, and the same happenswhen a y = 0, by performing a cyclic permutation on the y indices.The number of situations where one of i, a, b consists of equal indices is: K = M p + r N p − ( M p − M )( M r − M )( N p − N )By dividing by M p + r N p , we obtain the formula in the statement.The assertions about M = 1 , N = 1 , p = 1 , r = 1 are clear, because in all these casesthe product in the definition of α rp ( M, N ) vanishes, and so α rp ( M, N ) = 1.Finally, at p = 2, the equations defining d r ( M, N ) are as follows:[( i x + a , b ) , ( i x + a , b ) , ( i x +1 + a , b ) , ( i x +1 + a , b )]= [( i x + a , b ) , ( i x + a , b ) , ( i x +1 + a , b ) , ( i x +1 + a , b )] , ∀ x We already know that these conditions are satisfied when a = a or b = b . So,assume a = a , b = b . The element ( i x + a , b ) must appear somewhere at right, andthe only possible choice is ( i x + a , b ) = ( i x +1 + a , b ), which gives i x = i x +1 . Thus, allthe i x indices must be are equal, and we are done. (cid:3) In general, the situation is more complicated. As a first remark, we have:
Proposition 3.4.
We have d rp ( M, N ) ≥ δ p ( M, N ) , where δ p ( M, N ) = 1(
M N ) p (cid:26) a , . . . , a p ∈ { , . . . , M − } b , . . . , b p ∈ { , . . . , N − } (cid:12)(cid:12)(cid:12) [( a , b ) , ( a , b ) , . . . , ( a p , b p )]= [( a , b p ) , ( a , b ) , . . . , ( a p , b p − )] (cid:27) where the sets between brackets are as usual sets with repetitions.Proof. This is indeed clear from the fact that the conditions defining δ rp ( M, N ) are triviallysatisfied when the indices a, b satisfy [( a y , b y )] = [( a y , b y +1 )]. (cid:3) We can merge and extend Proposition 3.3 and Proposition 3.4, as follows:
Theorem 3.5.
When i consists of equal indices, or when [( a y , b y )] = [( a y , b y +1 )] , theconditions defining d rp ( M, N ) are trivially satisfied. The corresponding contribution is β rp ( M, N ) = δ p ( M, N ) + 1 M r − (1 − δ p ( M, N )) and this quantity equals d rp ( M, N ) at M = 1 , N = 1 , r = 1 , or p ≤ . EFORMED FOURIER MODELS WITH FORMAL PARAMETERS 9
Proof.
The first assertion is clear, and by definition of δ p ( M, N ), the corresponding con-tribution is the one in the statement. Since at M = 1, N = 1, r = 1 or p ≤ β rp ( M, N ) = α rp ( M, N ), the results here follow from Proposition 3.3.It remains to discuss the case p = 3. Here the equations are as follows:[( i x + a , b ) , ( i x + a , b ) , ( i x + a , b ) , ( i x +1 + a , b ) , ( i x +1 + a , b ) , ( i x +1 + a , b )]= [( i x + a , b ) , ( i x + a , b ) , ( i x + a , b ) , ( i x +1 + a , b ) , ( i x +1 + a , b ) , ( i x +1 + a , b )]We must prove that all the solutions are trivial, in the sense that either all the i x areequal, or the following condition is satisfied:[( a , b ) , ( a , b ) , ( a , b )] = [( a , b ) , ( a , b ) , ( a , b )]So, assume that we are in the non-trivial case, and pick x such that i x = i x +1 . Letus look now at the first element appearing on the left in the above equation, namely( i + a , b ). Since this element must appear as well on the right, we have 6 cases to beinvestigated. Observe now that in these 6 cases we must have, respectively: b = b , b = b , a = a , i x = i x +1 , b = b , b = b Thus, we have one case which is impossible, namely the one needing i x = i x +1 , and inthe other 5 cases, we always obtain a relation of type a i = a j or b i = b j , with i = j .So, assume a i = a j , with i = j . By using a cyclic permutation of the indices, we canassume that we have a = a . Now observe that our equations simplify, as follows:[( i x + a , b ) , ( i x + a , b ) , ( i x + a , b ) , ( i x +1 + a , b ) , ( i x +1 + a , b ) , ( i x +1 + a , b )]= [( i x + a , b ) , ( i x + a , b ) , ( i x + a , b ) , ( i x +1 + a , b ) , ( i x +1 + a , b ) , ( i x +1 + a , b )]As for the condition [( a y , b y )] = [( a y , b y +1 )], this simplifies as well, as follows:[( a , b ) , ( a , b ) , ( a , b )] = [( a , b ) , ( a , b ) , ( a , b )]Summarizing, the simplifications make dissapear the variables a , b , and so we are ledto a p = 2 problem, where the solutions are already known to be trivial.In the case b i = b j , with i = j , the situation is similar. By cyclic permutation we canassume b = b , and our equations simplify, as follows:[( i x + a , b ) , ( i x + a , b ) , ( i x + a , b ) , ( i x +1 + a , b ) , ( i x +1 + a , b ) , ( i x +1 + a , b )]= [( i x + a , b ) , ( i x + a , b ) , ( i x + a , b ) , ( i x +1 + a , b ) , ( i x +1 + a , b ) , ( i x +1 + a , b )]As for the condition [( a y , b y )] = [( a y , b y +1 )], this simplifies as well, as follows:[( a , b ) , ( a , b ) , ( a , b )] = [( a , b ) , ( a , b ) , ( a , b )]Thus, we are led once again to a p = 2 problem, whose solutions are trivial. (cid:3) Higher truncations
We know from Theorem 3.5 above that at small values of the truncation parameter,namely p = 1 , ,
3, the numbers d rp ( M, N ) come only from “trivial contributions”.At p = 4 and higher the situation becomes considerably more complex, involving thearithmetics of M, N , and this even in the simplest case, r = 2.We have here the following result, that we won’t use in what follows, but which mightbe interesting for instance in connection with the speculations in [1]: Theorem 4.1.
We have the formula d ( M, N ) = β ( M, N ) + δ | M ( M − N − M N where δ | M ∈ { , } is equal to when M is odd, and to when M is even.Proof. We have two equations, the one at x = 1 being as follows:[( i + a , b ) , . . . , ( i + a , b ) , ( i + a , b ) , . . . , ( i + a , b )]= [( i + a , b ) , . . . , ( i + a , b ) , ( i + a , b ) , . . . , ( i + a , b )]As for the equation at x = 2, this is as follows:[( i + a , b ) , . . . , ( i + a , b ) , ( i + a , b ) , . . . , ( i + a , b )]= [( i + a , b ) , . . . , ( i + a , b ) , ( i + a , b ) , . . . , ( i + a , b )]Since these equations are equivalent, we are left with the x = 1 equation.In order to compute the non-trivial contributions, we can assume i = i . Let us lookat the first element appearing on the left, ( i + a , b ). Since this element must appear aswell on the right, we have 8 cases to be investigated. In these 8 cases, we must have: b = b , a = a , b = b , a = a , i = i , b = b , ( i + a , b ) = ( i + a , b ) , b = b Thus one case is impossible, 6 cases reduce to the case p = 3, by using a cyclic reduction,as in the proof of Theorem 3.5, and there is one case left, ( i + a , b ) = ( i + a , b ).The same argument applies to the other 7 elements appearing on the left, and weconclude that the non-trivial solutions could only come from:( i + a x , b x ) = ( i + a x +2 , b x +2 ) , ( i + a x , b x +1 ) = ( i + a x +2 , b x +3 )Thus our indices i, a, b must be of the following special form, with 2 i = 0: i = ( i , i + i ) a = ( a , a , i + a , i + a ) b = ( b , b , b , b )In order to find now the non-trivial solutions, we must assume that we have i = 0, and[( a y , b y )] = [( a y , b y +1 )]. But, by translating by i , this latter condition reads:[( a , b ) , ( a , b ) , ( i + a , b ) , ( i + a , b )] = [( a , b ) , ( a , b ) , ( i + a , b ) , ( i + a , b )] EFORMED FOURIER MODELS WITH FORMAL PARAMETERS 11
Thus we must have b = b , and a = a , a = i + a as well.We can now compute the non-trivial contribution. This is given by: K = 1 M N · M δ | M · M ( M − · N ( N − M N is the normalization factor from the definition of d ( M, N ),then
M δ | M comes from the choice of i and of i = 0 satisfying 2 i = 0, then M ( M − a and of a = a , i + a , and finally N ( N −
1) comes from thechoice of b = b . But this gives the formula in the statement, and we are done. (cid:3) As a conclusion, the exact computation of d rp ( M, N ) is an interesting problem. In whatfollows we will only study the asymptotics of these numbers, with the result that theestimate d rp ( M, N ) ≥ β rp ( M, N ) from Theorem 3.5 becomes an equality, with r → ∞ .5. Limiting moments
Let us go back now to the numbers δ p ( M, N ), from Proposition 3.4 above.These numbers are known since [4] to be the rescaled moments of the main character forthe matrix model associated to F G × H ( Q ), where | G | = M, | H | = N , and where Q ∈ T G × H is generic. We will prove now that our moments are precisely these numbers:lim r →∞ d rp ( M, N ) = δ p ( M, N )For this purpose, observe that both d rp ( M, N ) , δ p ( M, N ) count, modulo some normal-izations, the solutions of certain equations on the indices a , . . . , a p ∈ { , . . . , M − } and b , . . . , b p ∈ { , . . . , N − } . We will prove the convergence componentwise, with respectto these pairs of multi-indices ( a, b ). We use the following simple fact: Proposition 5.1.
We have [ a y ] = [ b y ] inside a finite abelian group G precisely when X y χ ( a y ) = X y χ ( b y ) as an equality of complex numbers, for any character χ ∈ b G .Proof. By linearity, we have the following equivalences:[ a y ] = [ b y ] ⇐⇒ X y a y = X y b y inside C ∗ ( G ) ⇐⇒ ϕ X y a y ! = ϕ X y b y ! , ∀ ϕ ∈ C ( G ) ⇐⇒ χ X y a y ! = χ X y b y ! , ∀ χ ∈ b G Thus, we obtain the condition in the statement. (cid:3)
Now back to our question, since only the cardinalities M = | G | , N = | H | are revelant,we can assume G = Z M , H = Z N . We first have the following technical result: Proposition 5.2.
For a pair of multi-indices ( a, b ) , the following are equivalent: (1) [( a y , b y )] = [( a y , b y +1 )] . (2) [( i + a y , b y ) , ( a y , b y +1 )] = [( i + a y , b y +1 ) , ( a y , b y )] , for any i ∈ Z M .Proof. Observe that (1) = ⇒ (2) is clear. For (2) = ⇒ (1), we use Proposition 5.1. Byusing the identification \Z M × Z N ≃ d Z M × c Z N , we have, with η ∈ d Z M , ρ ∈ c Z N :[( i + a y , b y ) , ( a y , b y +1 )] = [( i + a y , b y +1 ) , ( a y , b y )] , ∀ i ⇐⇒ X y η ( i + a y ) ρ ( b y ) + η ( a y ) ρ ( b y +1 ) = X y η ( i + a y ) ρ ( b y +1 ) + η ( a y ) ρ ( b y ) , ∀ i, η, ρ ⇐⇒ η ( i ) X y η ( a y ) ρ ( b y ) − η ( a y ) ρ ( b y +1 ) = X y η ( a y ) ρ ( b y ) − η ( a y ) ρ ( b y +1 ) , ∀ i, η, ρ ⇐⇒ X y η ( a y ) ρ ( b y ) − η ( a y ) ρ ( b y +1 ) = 0 , ∀ η, ρ ⇐⇒ [( a y , b y )] = [( a y , b y +1 )]Thus, we have obtained the equivalence in the statement. (cid:3) With the above result in hand, we can prove the estimate that we need, namely:
Proposition 5.3.
Assuming [( a y , b y )] = [( a y , b y +1 )] , the number K rp ( a, b ) = 1 M r (cid:26) i , . . . , i r ≤ M (cid:12)(cid:12)(cid:12) [( i x + a y , b y ) , ( i x +1 + a y , b y +1 )]= [( i x + a y , b y +1 ) , ( i x +1 + a y , b y )] , ∀ x (cid:27) goes to in the r → ∞ limit.Proof. Observe that the problem is already solved at p ≤
3, because by Theorem 3.5 allthe i x indices must be equal, and so the number in the statement is: K r ( a, b ) = 1 M r − → S ⊂ { , . . . , M − } consisting of the solutions i of thefollowing equation: [( i + a y , b y ) , ( a y , b y +1 )] = [( i + a y , b y +1 ) , ( a y , b y )]In terms of this set, the quantity in the statement is given by: K rp ( a, b ) = 1 M r n i , . . . , i r ≤ M (cid:12)(cid:12)(cid:12) i − i , . . . , i r − i ∈ S o Now by ignoring the last condition, we have M choices for i , then | S | choices for i , | S | choices for i , and so on, up to | S | choices for i r . Thus, we obtain: K rp ( a, b ) ≤ M r · M · | S | · . . . | S | = (cid:18) | S | M (cid:19) r − EFORMED FOURIER MODELS WITH FORMAL PARAMETERS 13
On the other hand, by Proposition 5.2 our assumption [( a y , b y )] = [( a y , b y +1 )] implies S = { , . . . , M − } . In particular we have | S | ≤ M −
1, and this gives the result. (cid:3)
With the above estimate in hand, we can now prove:
Theorem 5.4.
We have the formula lim r →∞ d rp ( M, N ) = δ p ( M, N ) valid for any p ≥ and any M, N ∈ N .Proof. Our claim is that we have, for any pair of multi-indices ( a, b ):lim r →∞ K rp ( a, b ) = δ [( a y ,b y )] , [( a y ,b y +1 )] Indeed, when [( a y , b y )] = [( a y , b y +1 )], this is exactly what we found in Proposition 5.3.As for the remaining case [( a y , b y )] = [( a y , b y +1 )], this is trivial, because here the equationsdefining K rp ( a, b ) are all trivial, and so we have K rp ( a, b ) = 1, for any r ∈ N . (cid:3) Summarizing, we have proved that the law of the main character for F G,H coincideswith that computed in [4], for the matrix F G × H ( Q ), with Q ∈ T G × H generic. As aconsequence, all the findings in [4] apply. In what follows we will review these results, byusing an analytic approach, and by bringing some technical improvements.6. Gram matrices
We study now the behavior of the limiting moments δ p ( M, N ) that we found, in the p → ∞ limit. For this purpose, let us first recall the following result, from [4]: Proposition 6.1.
We have the formula δ p ( M, N ) = 1(
M N ) p Z T MN T r ( G ( Q ) p ) dQ where G ∈ M M ( C ( T MN )) is given by G ( Q ) = Gram matrix of the rows of Q .Proof. If we denote by R , . . . , R M ∈ T N the rows of Q ∈ T MN , we have: δ p ( M, N ) = 1(
M N ) p X a ...a p X b ...b p δ [ a b ,...,a p b p ] , [ a b p ,...,a p b p − ] = 1( M N ) p Z T MN X a ...a p X b ...b p Q a b . . . Q a p b p Q a b p . . . Q a p b p − dQ = 1( M N ) p Z T MN X a ...a p < R a , R a >< R a , R a > . . . < R a p , R a > dQ But this gives the formula in the statement, and we are done. (cid:3)
In the case M = 2 some simplifications appear, and we have: Proposition 6.2.
We have the formula δ p (2 , N ) = 12 p − X k ≥ (cid:18) p k (cid:19) Z T N (cid:12)(cid:12)(cid:12)(cid:12) q + . . . + q N N (cid:12)(cid:12)(cid:12)(cid:12) k dq with the integral at right being with respect to the uniform measure on T N .Proof. We use the formula in Proposition 6.1. If we denote by R , R ∈ T N the rows of Q then, with q = R /R ∈ T N , the Gram matrix that we are interested in is: G ( Q ) = (cid:18) N q + . . . + q N ¯ q + . . . + ¯ q N N (cid:19) Thus, with S = ( q + . . . + q N ) /N , we have G ( Q ) = N A ( q ), where: A ( q ) = (cid:18) S ¯ S (cid:19) Now since q ∈ T N is uniform when Q ∈ T N is uniform, we deduce that we have: δ p (2 , N ) = 12 p Z T N X a ...a p A ( q ) a a A ( q ) a a . . . A ( q ) a p a dq The point now is that the nontrivial factors in the above product, namely S, ¯ S , willform together | S | k factors, with k ≥
0. To be more precise, in order to find the numberof | S | k summands, we have to count the circular configurations consisting of p numbers1 ,
2, such that both the 1 values and the 2 values are arranged into k non-empty intervals.By looking at the endpoints of these 2 k intervals, we have 2 (cid:0) p k (cid:1) choices, so the k -thcontribution is C k = 2 (cid:0) k p (cid:1) | S | k . Thus, we have the following formula: δ p (2 , N ) = 12 p X k ≥ (cid:18) p k (cid:19) Z T N | S | k dq But this gives the formula in the statement, and we are done. (cid:3)
We write a k ≃ b k when a k /b k →
1. We will need the following result, due to Richmondand Shallit [19]:
Proposition 6.3.
We have the estimate Z T N (cid:12)(cid:12)(cid:12)(cid:12) q + . . . + q N N (cid:12)(cid:12)(cid:12)(cid:12) k dq ≃ s N N (4 πk ) N − valid in the k → ∞ limit. EFORMED FOURIER MODELS WITH FORMAL PARAMETERS 15
Proof.
This is a reformulation of the result in [19]. Observe first that we have: Z T N (cid:12)(cid:12)(cid:12) q + . . . + q N (cid:12)(cid:12)(cid:12) k dq = Z T N X i ...i k X j ...j k q i . . . q i k q j . . . q j k dq = (cid:26) i . . . i k ∈ { , . . . , N − } j . . . j k ∈ { , . . . , N − } (cid:12)(cid:12)(cid:12) [ i , . . . , i k ]= [ j , . . . , j k ] (cid:27) Let us examine now the numbers on the right. If we denote by r , . . . , r N the numberof occurrences of 0 , . . . , N − i ] = [ j ], then r + . . . + r N = k ,and the corresponding solutions of [ i ] = [ j ] come by dividing, once for i , and once for j ,the set { , . . . , k } into subsets of size r , . . . , r N . Thus, we have: Z T N (cid:12)(cid:12)(cid:12) q + . . . + q N (cid:12)(cid:12)(cid:12) k dq = X k =Σ r i (cid:18) kr , . . . , r N (cid:19) By using now the estimate in [19], we obtain the result. (cid:3)
We can now deduce a final estimate at M = 2, as follows: Theorem 6.4.
We have the estimate δ p (2 , N ) ≃ s N N ( πp ) N − valid in the p → ∞ limit.Proof. We use the formula in Proposition 6.2. Since for any
T > k < T won’tcontribute to the p → ∞ limit, we can use Proposition 6.3, and we obtain: δ p (2 , N ) ≃ s N N (2 π ) N − · p − X k ≥ (cid:18) p k (cid:19) p (2 k ) N − Let us denote by A even the average of 2 p − terms on the right. This average is indexed bythe integers s = 2 k in an obvious way, and we can consider as well the “complementary”quantity A odd , indexed by the integers s = 2 k + 1. By estimating | A even − A odd | we deducethat we have A even ≃ A odd , and so A even ≃ A even + A odd . Thus, we have: δ p (2 , N ) ≃ s N N (2 π ) N − · p X s ≥ (cid:18) ps (cid:19) √ s N − On the other hand, by derivating several times the binomial formula (1 + x ) p = P s ≥ (cid:0) ps (cid:1) x s , and then evaluating at x = 1, we have the following estimate:12 p X s ≥ (cid:18) ps (cid:19) s α ≃ (cid:16) p (cid:17) α With α = (1 − N ) /
2, this gives the following formula: δ p (2 , N ) ≃ s N N (2 π ) N − · s(cid:18) p (cid:19) N − But this gives the formula in the statement, and we are done. (cid:3) Partition decomposition
Our purpose now will be that of estimating δ p ( M, N ), when
M, N ∈ N are arbitrary.The idea will be that of decomposing over partitions. First, we have: Proposition 7.1.
We have the formula δ p ( M, N ) = 1(
M N ) p X π⊲σ M !( M − | π | )! · N !( N − | σ | )! where for π, σ ∈ P ( p ) we write π ⊲ σ when | β ∩ γ | = | ( β − ∩ γ | , ∀ β ∈ π, ∀ γ ∈ σ .Proof. We know that δ p ( M, N ) is the probability for [( a x , b x )] = [( a x , b x +1 )] to happen.We can split this quantity over pairs of partitions, as follows: δ p ( M, N ) = 1(
M N ) p X π,σ ∈ P ( p ) (cid:26) a , . . . , a p ∈ { , . . . , M − } b , . . . , b p ∈ { , . . . , N − } (cid:12)(cid:12)(cid:12) ker a = π, ker b = σ [( a x , b x )] = [( a x , b x +1 )] (cid:27) Now observe that the validity of the condition [( a x , b x )] = [( a x , b x +1 )] depends only onthe partitions π = ker a, σ = ker b . To be more precise, this condition is satisfied preciselywhen the condition π ⊲ σ in the statement holds. We therefore obtain: δ p ( M, N ) = 1(
M N ) p X π⊲σ (cid:26) a , . . . , a p ∈ { , . . . , M − } b , . . . , b p ∈ { , . . . , N − } (cid:12)(cid:12)(cid:12) ker a = π ker b = σ (cid:27) But this gives the formula in the statement, and we are done. (cid:3)
As an application, we can discuss what happens in the M = tN → ∞ regime, whichmeans N → ∞ and M = tN + o (1), with t > Proposition 7.2.
With M = tN → ∞ we have δ p ( M, N ) ≃ S p ( t ) M − p N where S p ( t ) = P π ∈ NC ( p ) t | π | is the Stirling polynomial of N C ( p ) .Proof. According to the formula in Proposition 7.1, with M = tN → ∞ we have: δ p ( M, N ) ≃ X π⊲σ M | π |− p N | σ |− p EFORMED FOURIER MODELS WITH FORMAL PARAMETERS 17
We use now the standard fact that π ⊲ σ implies | π | + | σ | ≤ p + 1, with equality when π, σ ∈ N C ( p ) are inverse to each other, via Kreweras complementation. We obtain: δ p ( M, N ) ≃ X π ∈ NC ( p ) M | π |− p N −| π | But this gives the formula in the statement, and we are done. See [4]. (cid:3)
Now back to our original question, concerning the case where
M, N ∈ N are fixed, wecan rewrite the formula in Proposition 7.1 in a more convenient way, as follows: Proposition 7.3.
We have the formula δ p ( M, N ) = M X s =1 N X t =1 M !( M − s )! · S ps M p · N !( N − t )! · S pt N p · P (cid:16) π ⊲ σ (cid:12)(cid:12)(cid:12) | π | = s, | σ | = t (cid:17) where S ps = { π ∈ P ( p ) || π | = s } are the Stirling numbers of P ( p ) .Proof. According to the formula in Proposition 7.1, we have: δ p ( M, N ) = 1(
M N ) p M X s =1 N X t =1 M !( M − s )! · N !( N − t )! (cid:16) π ⊲ σ (cid:12)(cid:12)(cid:12) | π | = s, | σ | = t (cid:17) On the other hand, the probability in the statement is given by: P (cid:16) π ⊲ σ (cid:12)(cid:12)(cid:12) | π | = s, | σ | = t (cid:17) = (cid:16) π ⊲ σ (cid:12)(cid:12)(cid:12) | π | = s, | σ | = t (cid:17) S ps S pt By combining these two formulae, we obtain the result. (cid:3)
Consider the probabilities which appear on the right in Proposition 7.3: ε p ( s, t ) = P (cid:16) π ⊲ σ (cid:12)(cid:12)(cid:12) | π | = s, | σ | = t (cid:17) The corresponding contributions to δ p ( M, N ) are then given by: δ stp ( M, N ) = M !( M − s )! · S ps M p · N !( N − t )! · S pt N p · ε p ( s, t )The idea now will be to separate the contributions coming from indices s = 1 or t = 1.To be more precise, we can rewrite Proposition 7.3 as follows: Theorem 7.4.
We have the formula δ p ( M, N ) = 1 M p − + 1 N p − − M N ) p − + M X s =2 N X t =2 δ stp ( M, N ) where δ stp ( M, N ) are the contributions defined above. Proof.
According to Proposition 7.3, we have the following formula: δ p ( M, N ) = M X s =1 N X t =1 δ stp ( M, N )Since we have ε p (1 , t ) = 1, the contributions at s = 1 are given by: δ tp ( M, N ) = M · M p · N !( N − t )! · S pt N p = 1 M p − · N !( N − t )! · S pt N p Now by summing over t ≥
1, we obtain the following formula: N X t =1 δ tp ( M, N ) = 1 M p − N X t =1 N !( N − t )! · S pt N p = 1 M p − Similarly, we have as well the following formula: M X s =1 δ s p ( M, N ) = 1 N p − M X s =1 M !( M − s )! · S ps M p = 1 N p − Finally, at s = 1 , t = 1 the contribution is as follows: δ p ( M, N ) = M · M p · N · N p = 1( M N ) p − By using the inclusion-exclusion principle, this gives the result. (cid:3) Moment estimates
In this section we estimate δ p ( M, N ), by using the formula found in Theorem 7.4. Inorder to deal with the contributions at s ≥ , t ≥
2, we use the following fact:
Proposition 8.1.
The function constructed above, ε p ( s, t ) = P (cid:16) π ⊲ σ (cid:12)(cid:12)(cid:12) | π | = s, | σ | = t (cid:17) is decreasing in both s ∈ N and t ∈ N .Proof. The problem being symmetric in s, t , it is enough to prove that ε p ( s, t ) is decreasingin t . By splitting the problem over the partitions π satisfying | π | = s , it is enough toprove that for any partition π ∈ P ( p ), the following quantity is decreasing with t : ε π ( t ) = P (cid:16) π ⊲ σ (cid:12)(cid:12)(cid:12) | σ | = t (cid:17) In order to do so, recall from Proposition 7.1 that π ⊲ σ is equivalent to: | β ∩ γ | = | ( β − ∩ γ | , ∀ β ∈ π, ∀ γ ∈ σ Now observe that when merging two blocks of σ , say ( γ , γ ) → γ , the condition issatisfied for γ , simply by summing the equalities for γ , γ . We deduce from this that theprobability ε π ( t ) gets bigger when decreasing the number t = | σ | , as desired. (cid:3) EFORMED FOURIER MODELS WITH FORMAL PARAMETERS 19
Let us combine now Theorem 7.4 with Proposition 8.1. We obtain:
Proposition 8.2.
We have the estimate δ p ( M, N ) ≤ − (cid:18) − M p − (cid:19) (cid:18) − N p − (cid:19) (cid:0) − ε p (2 , (cid:1) valid for any M, N ≥ .Proof. The formula in Theorem 7.4 above can be written as follows: δ p ( M, N ) = 1 M p − + 1 N p − − M N ) p − + M X s =2 N X t =2 δ stp ( M, N )= 1 − (cid:18) − M p − (cid:19) (cid:18) − N p − (cid:19) + M X s =2 N X t =2 δ stp ( M, N )According now to Proposition 8.1, for any s, t ≥ δ stp ( M, N ) ≤ M !( M − s )! · S ps M p · N !( N − t )! · S pt N p · ε p (2 , s, t ≥
2, and by using the inclusion-exclusion principle,as in the proof of Theorem 7.4, we obtain: M X s =2 N X t =2 δ stp ( M, N ) ≤ M X s =2 N X t =2 M !( M − s )! · S ps M p · N !( N − t )! · S pt N p · ε p (2 , (cid:18) − M p − − N p − + 1( M N ) p − (cid:19) ε p (2 , (cid:18) − M p − (cid:19) (cid:18) − N p − (cid:19) ε p (2 , (cid:3) On the other hand, by using the results obtained in section 6 above, we have:
Proposition 8.3.
We have the estimate ε p (2 , N ) ≃ · N ! s N N ( πp ) N − valid in the p → ∞ limit. Proof.
We have the following estimate, in the p → ∞ limit: δ stp ( M, N ) = M !( M − s )! · S ps M p · N !( N − t )! · S pt N p · ε p ( s, t ) ≃ M !( M − s )! · s p M p · N !( N − t )! · t p N p · ε p ( s, t )= M !( M − s )! · N !( N − t )! (cid:18) stM N (cid:19) p ε p ( s, t )Here we have used the estimate S ps ≃ s p , which follows from the fact that choosing apartition π ∈ P ( p ) with ≤ s blocks amounts in assigning a number 1 , . . . , s to any of thepoints 1 , . . . , p , and the assignements which lead to | π | < s can be neglected.In particular, at s = M = 2 we obtain: δ tp (2 , N ) ≃ · N !( N − t )! (cid:18) tN (cid:19) p ε p (2 , t )By combining this estimate with Theorem 7.4 at M = 2, we obtain: δ p (2 , N ) = 12 p − + 1 N p − − N ) p − + N X t =2 δ tp (2 , N ) ≃ p − + 2 N X t =2 N !( N − t )! (cid:18) tN (cid:19) p ε p (2 , t )With this formula in hand, we can proceed by recurrence on N ≥
2. Since the quantityin the statement converges with p → ∞ to 0 much slower than the various powers α N ,with α ∈ (0 , δ p (2 , N ) ≃ · N ! ε p (2 , N )Now by using the M = 2 estimate from Theorem 6.4, we obtain: ε p (2 , N ) ≃ · N ! · δ p (2 , N ) ≃ · N ! s N N ( πp ) N − Thus we have obtained the formula in the statement. (cid:3)
With the above results in hand, we can now prove our result:
Theorem 8.4.
We have lim p →∞ δ p ( M, N ) = 0 , for any
M, N ≥ .Proof. By combining Proposition 8.2 and Proposition 8.3, we obtain: δ p ( M, N ) ≤ − (cid:18) − M p − (cid:19) (cid:18) − N p − (cid:19) (cid:0) − ε p (2 , (cid:1) Since the product on the right converges to 1 × × (cid:3) EFORMED FOURIER MODELS WITH FORMAL PARAMETERS 21 Poisson laws
We recall that the free analogue of the Poisson law of parameter t >
0, in the sense ofthe Bercovici-Pata bijection [9], is the Marchenko-Pastur law of parameter t , also calledfree Poisson law of parameter t . We denote this measure by π t . See [16], [17], [22].We have the following result, summarizing our findings: Theorem 9.1.
Given two finite abelian groups
G, H , with | G | = M, | H | = N , considerthe main character χ of the quantum group associated to F G × H . (1) µ = law ( χMN ) is supported on [0 , . (2) This measure µ has no atom at . (3) With M = tN → ∞ we have law (cid:0) χN (cid:1) = (cid:0) − M (cid:1) δ + M π t , in moments.Proof. In this statement (1) is trivial, (2) is new, and (3) is since known since [4], in thecase of the generic fibers. To be more precise, the proof goes as follows:(1) This follows from the fact that χ is by definition the main character for a certainquantum group G ⊂ S + MN , and is therefore a sum of M N projections.(2) This follows from Theorem 8.4 above, and from the fact that an atom at 1 wouldmake the moments converge to a nonzero quantity.(3) According to our various normalizations, we have: Z r G (cid:16) χN (cid:17) p = c rp ( M, N ) N p = ( M N ) p − d rp ( M, N ) N p = M p − N d rp ( M, N )By using Proposition 7.2 we obtain, in the M = tN → ∞ limit: Z G (cid:16) χN (cid:17) p ≃ M p − N δ p ( M, N ) ≃ M p − N S p ( t ) M − p N = 1 M S p ( t )Now since S p ( t ) is the p -th moment of π t , this gives the result. (cid:3) Concluding remarks
There are several questions, in relation with the above results. First, we do not knowhow to improve Theorem 8.4, with a precise estimate, as in Theorem 6.4.There are as well some interesting questions in relation with [1], [21]. The main problemhere, well-known and open, is that of understanding how a general deformed Fouriermatrix F K can be defined, directly in terms of the finite abelian group K .In relation now with [7], observe that the representations there are as well of the form π : C ( S +dim B ) → C ( U B , L ( B )), for a certain finite dimensional C ∗ -algebra B . In thepresent paper this algebra is a commutative one, B = C ( G × H ). We believe that theunification with [7] is an important question, which could lead to a substantial “boost”in the understanding and use of the integration formula in [6], [24]. References [1] T. Banica, First order deformations of the Fourier matrix,
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